Group #20274796 — subgroups

Subgroups — #20274796

Order 6784.

272 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 85 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133176	C₂	1	1	yes	maximal, characteristic
2	#16810587	C₂	1	1	yes	maximal, characteristic
3	#16810391	C₂	1	4	yes	maximal, complemented
4	C₅₃×#407	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719917	C₂²	1	2	yes	—
6	#13133176	—	2	2	no	—
7	#13133176	—	2	2	no	—
8	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
9	C₂×C₂²⋊Dic₅₃	C₄	1	2	yes	—
10	C₂×C₄×Dic₅₃	C₄	1	2	yes	—
11	C₂²⋊#88322	—	2	2	no	—
12	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×M₁₆×C₅₃	C₂²	1	2	yes	—
14	C₂²⋊C₈×C₅₃	—	2	2	no	—
15	C₂⁴×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₅₃	—	2	4	no	—
17	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	D₄	1	2	yes	—
19	C₂²×C₄×C₅₃	D₄	1	2	yes	—
20	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
21	C₂²×Dic₅₃	C₂×C₄	1	4	yes	—
22	M₁₆×C₅₃	—	2	4	no	—
23	C₂×C₈×C₅₃	—	2	2	no	—
24	C₂²⋊Dic₅₃	—	2	4	no	—
25	C₄⋊Dic₅₃	—	2	2	no	—
26	C₄⋊Dic₅₃	—	2	2	no	—
27	C₄×Dic₅₃	—	2	4	no	—
28	#11719917	—	2	4	no	—
29	C₂×#88322	—	2	2	no	—
30	C₂³×C₅₃	—	2	2	no	—
31	C₂³×C₅₃	—	4	2	no	—
32	C₂³×C₅₃	C₂²⋊C₄	1	2	yes	—
33	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	—	2	2	no	—
35	C₂×C₄×C₅₃	—	2	4	no	—
36	C₂×C₄×C₅₃	—	2	4	no	—
37	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₅₃	#41	1	4	yes	—
41	C₂×Dic₅₃	—	2	4	no	—
42	C₂×Dic₅₃	—	4	4	no	—
43	C₈×C₅₃	—	4	2	no	—
44	#88322	—	4	2	no	—
45	C₂²×C₅₃	—	2	2	no	—
46	C₂²×C₅₃	—	2	2	no	—
47	C₂²×C₅₃	—	2	4	no	—
48	C₂²×C₅₃	—	4	4	no	—
49	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₅₃	#117	1	2	yes	—
51	C₄×C₅₃	—	2	2	no	—
52	C₄×C₅₃	#123	1	2	yes	—
53	C₄×C₅₃	#123	1	2	yes	—
54	Dic₅₃	—	4	4	no	—
55	#1822	—	53	1	no	maximal, Sylow
56	C₂×C₅₃	—	2	2	no	—
57	C₂×C₅₃	—	4	2	no	—
58	C₂×C₅₃	#381	1	1	yes	characteristic
59	C₂×C₅₃	#413	1	2	yes	—
60	C₂×#117	—	53	1	no	—
61	#407	—	53	1	no	—
62	#407	D₅₃	1	1	yes	characteristic
63	#343	—	53	4	no	—
64	C₅₃	#1822	1	1	yes	characteristic, complemented, Sylow
65	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
66	C₂×M₁₆	—	53	2	no	—
67	C₂×M₁₆	C₂×D₅₃	1	2	yes	—
68	#117	—	106	2	no	—
69	#117	—	106	2	no	—
70	C₂×C₄⋊C₄	—	53	2	no	—
71	C₂×C₂²⋊C₄	—	53	2	no	—
72	C₂×C₄²	—	53	2	no	—
73	C₂²⋊C₈	—	2	2	no	—
74	C₂²⋊C₈	—	106	2	no	—
75	C₂⁴	C₄×D₅₃	1	1	yes	characteristic
76	C₂²×C₄	—	2	4	no	—
77	C₂²×C₄	—	53	4	no	—
78	C₂²×C₄	C₄×D₅₃	1	1	yes	characteristic
79	C₂²×C₄	D₂₁₂	1	2	yes	—
80	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
81	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
82	M₁₆	—	2	4	no	—
83	M₁₆	—	106	4	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	106	2	no	—
86	C₄⋊C₄	—	106	2	no	—
87	C₄⋊C₄	—	106	2	no	—
88	C₂²⋊C₄	—	106	4	no	—
89	C₄²	—	106	4	no	—
90	C₂³	—	2	2	no	—
91	C₂³	—	4	2	no	—
92	C₂³	#11719921	1	2	yes	—
93	C₂³	C₂×C₄×D₅₃	1	1	yes	characteristic
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	106	4	no	—
98	C₂×C₄	—	212	4	no	—
99	C₂×C₄	#11719921	1	2	yes	—
100	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
101	C₂×C₄	#11719947	1	4	yes	—
102	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
103	C₈	—	4	2	no	—
104	C₈	—	212	2	no	—
105	C₂²	—	2	2	no	—
106	C₂²	—	2	4	no	—
107	C₂²	—	2	2	no	—
108	C₂²	—	4	4	no	—
109	C₂²	C₂×#11719921	1	1	yes	characteristic
110	C₂²	#13133173	1	2	yes	—
111	C₄	—	2	2	no	—
112	C₄	—	212	4	no	—
113	C₄	#13133198	1	2	yes	—
114	C₄	#13133201	1	2	yes	—
115	C₂	—	2	2	no	—
116	C₂	—	4	2	no	—
117	C₂	#16810484	1	1	yes	characteristic
118	C₂	#16810605	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133176

C₂

yes

maximal, characteristic

#16810587

C₂

yes

maximal, characteristic

#16810391

C₂

yes

maximal, complemented

C₅₃×#407

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

#13133176

—

#13133176

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

C₂×C₂²⋊Dic₅₃

C₄

yes

—

C₂×C₄×Dic₅₃

C₄

yes

—

C₂²⋊#88322

—

C₂³×C₄×C₅₃

C₂²

yes

characteristic

C₂×M₁₆×C₅₃

C₂²

yes

—

C₂²⋊C₈×C₅₃

—

C₂⁴×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

#11719917

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²⋊C₄

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

#117

yes

—

C₄×C₅₃

—

C₄×C₅₃

#123

yes

—

C₄×C₅₃

#123

yes

—

Dic₅₃

—

#1822

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#381

yes

characteristic

C₂×C₅₃

#413

yes

—

C₂×#117

—

#407

—

#407

D₅₃

yes

characteristic

#343

—

C₅₃

#1822

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₅₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

—

#117

—

106

—

#117

—

106

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂²⋊C₈

—

C₂²⋊C₈

—

106

—

C₂⁴

C₄×D₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₅₃

yes

characteristic

C₂²×C₄

D₂₁₂

yes

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

106

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

106

—

C₂²⋊C₄

—

106

—

C₄²

—

106

—

C₂³

—

C₂³

—

C₂³

#11719921

yes

—

C₂³

C₂×C₄×D₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

#11719921

yes

—

100

C₂×C₄

C₂×D₂₁₂

yes

characteristic

101

C₂×C₄

#11719947

yes

—

102

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

103

C₈

—

104

C₈

—

212

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

C₂×#11719921

yes

characteristic

110

C₂²

#13133173

yes

—

111

C₄

—

112

C₄

—

212

—

113

C₄

#13133198

yes

—

114

C₄

#13133201

yes

—

115

C₂

—

116

C₂

—

117

C₂

#16810484

yes

characteristic

118

C₂

#16810605

yes

—

	subgroup	G/H
center	C₂²	C₂×#11719921
commutator	C₂³×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×#407	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	#407	D₅₃
hypocenter	C₅₃	#1822
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

subgroup

G/H

center

C₂²

C₂×#11719921

commutator

C₂³×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×#407

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #407

N	H ≅ G/N	type
3. #16810391	C₂	semidirect
64. C₅₃	#1822	semidirect

H ≅ G/N

type

3. #16810391

C₂

semidirect

64. C₅₃

#1822

semidirect

Subgroups — #20274796

Conjugacy classes of subgroups

Complements